Problem — Thought experiment, no displacement current

Problem

Imagine that the displacement-current term vanished from Maxwell’s equations. What would happen to the propagation of electromagnetic waves in vacuum? And to the night sky seen from Earth?

Solution

The role of the displacement current. The term Maxwell added to Ampère’s law is $\varepsilon_0\frac{d\Phi_E}{dt}$ and it is what allows a varying electric field to generate a magnetic field even where no real current flows (for example, in vacuum).

Without that term. If it vanished, a varying $\vv E$ would no longer generate a $\vv B$. Only Faraday’s law would remain (varying $\vv B$ $\to\vv E$), but the chain of mutual generation would break: one of the two links would be missing.

Consequence for waves. Without the mutual regeneration of the fields, no EM waves would propagate in vacuum: no light, no radio waves, no WiFi, no X-rays travelling through empty space.

The night sky. Starlight and light from galaxies crosses empty space precisely as an EM wave. Without propagation in vacuum, that light could not reach us: the cosmos would be invisible from Earth. The night sky would appear completely dark.

Summary $\varepsilon_0\,d\Phi_E/dt$ is the ingredient that closes the $\vv E\leftrightarrow\vv B$ cycle. Removing it means $\ev{\text{no EM waves in vacuum} \Rightarrow \text{dark sky}}$.

The displacement current

Connections

Topics: Electromagnetic waves Concepts: Displacement current · Electromagnetic wave Skills: Limiting-case and thought-experiment analysis